Preheat oven to 400. Combine flour, cornmeal, baking powder, baking soda and salt in a food processor fitted with a metal blade. Pulse several times to incorporate the ingredients. Add cheese and butter and pulse several more times, until the mixture looks pebbly with pea-sized lumps. Dump the mixture into a large bowl.

Add sour cream and chives to the mixture and stir it with a rubber spatula until just combined. Add 3 tablespoons of cream and stir until the dough just starts to come together. Add another tablespoon or two of cream, if needed. Mix the dough with your hands to help it come together, being careful not to overmix it.

On a lightly floured surface, gently pat the dough into a 9x5 inch rectangle that is a little more than ½ an inch thick. Using a biscuit cutter or sharp knife, cut into 12 biscuits.

Place the biscuits on a baking sheet and brush the tops lightly with some cream (this will help the tops brown). You can also sprinkle the tops with salt and freshly ground pepper if you'd like.

Bake the biscuits for 12-15 minutes, until lightly browned on the tops and bottoms. Serve warm.